Computation, Computers, and Programs Course Introduction

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CS20a Turing Machines (Oct 29, 2002)

• So far
• DFA regular languages
• PDA context-free languages
• Today
• Computability

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Churchs thesis

• The computable functions are the same as the
  partial recursive functions
• What is a computable function?
• Lots of models l-calculus, mechanical devices,
  fluids and valves, DNA, quantum devices
• All of these seem to define the same set of
  functions (but some have better performance)

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Definitions

• A problem is a yes-or-no question
• Are two CFGs equivalent?
• Does a TM halt on blank tape?
• An instance of a problem has specific arguments
• Does this TM halt on blank tape?
• An algorithm is a program that always halts (with
  the correct answer), so it is recursive
• A problem is decidable if it is recursive
• If not, it is undecidable
• Classically, every instance of a problem is
  decidable
• Not true constructively

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Some properties

• The complement of a recursive set is recursive
• Given M that halts on all inputs, construct M
  that simulates M.
• If M halts and accepts, M halts and does not
  accept
• If M halts and does not accept, M halts and
  accepts
• The union/intersection of two recursive sets is
  recursive
• Simulate machine M1, then simulate machine M2
• Union accept if either accepts
• Intersection accept if both accept
• If L and (Sigma - L) are r.e., then L is
  recursive

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Writing down a Turing Machine
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Universal Turing Machines
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Reals are uncountable

• Cantors diagonalization argument

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Another diagonalization argument
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Halting problem

• Problem determine entries in the matrix
• Find a recursive machine K, given Mx
• K halts and accepts if M accepts x
• K halts and rejects if M does not terminate
• Is there such a machine?
• Suppose so, then build a new machine N, that,
  given x, runs K on Mxx and
• Accepts if K rejects
• Loops forever of K accepts
• If K exists, then N does too let N My
• Ny halts and accepts if Ny does not halt
• Ny loops forever if Ny halts

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Halting problem
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Some undecidable problems

• Determining if M halts on blank tape
• To test if M halts on string x, build a machine
  that writes x onto the tape, then simulates M
• Determining whether L(M)
• To test if M halts on string x, build a machine
  that erases the input, writes x onto the tape,
  then simulates M (accept iff M terminates)
• Determining whether L(M) L(M)
• Let M be the machine that always halts and
  rejects

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Rices theorem

• A property P is a set of r.e. languages (so it is
  a set containing sets of strings)
• For any r.e. language L, we say P(L) is true iff
  L is a member of P
• Theorem Any nontrivial property of the r.e.
  languages is undecidable
• Nontrivial means the property is neither always
  true nor always false

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Proof of Rices theorem

• Consider a property P
• Assume P() false, otherwise invert P
• Since P is nontrivial, there is a language L in P
• Let ML be a TM accepting L
• To determine if M halts on input w, build M
• Ignore the input, and simulate M on w
• If M halts, then start ML on the input string
• So P(M) iff M halts on w

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Properties of TMs (not r.e. languages)

• Example 1 it is decidable if a TM has more than
  10 states
• Count them

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Properties of TMs (not r.e. languages)

• Example 2 it is undecidable if a TM every prints
  3 consecutive 0s
• For any M, construct a new machine M that
  represents 0 by 01, and 1 by 01
• Use M to simulate M (M never has 3 consecutive
  0s on the tape)
• Except if M halts, then M prints 3 zeros

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Properties of TMs (not r.e. languages)

• Example 3 it is decidable if a TM ever scans a
  tape cell 4 or more times
• If the TM never scan a cell 4 or more times, then
  each crossing sequence (where tape head move past
  the boundary between cells) happs at most 3 times
• But there is a finite number of crossing
  sequences of length 3 or less
• So the TM either stays within a fixed number of
  cells, or some crossing sequence repeats
• If some crossing sequence repeats, the tape head
  move right with a detectable pattern, and the
  problem is decidable

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TM executions

• Lets revisit the past
• This time with readable symbols

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Instantaneous descriptions
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Transitions
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Executions
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Valid computations (VALCOMPs)
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Context-free/VALCOMP
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CFG intersection
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Invalid computations
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Invalid computations
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CFGs
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Turing Machines

• TM are basically 2-way FA, with a writable tape
• The TM accept the r.e. languages
• TM that always halt accept the recursive
  languages
• Every known computational model can be mapped to
  TM (Churchs thesis)
• Nearly everything about TM is undecidable
• Wonderful!